Sword design would vary, at least in the French military, based on the type of unit. In the Revolutionary and Napoleonic army, there were three types of cavalry; light, medium, and heavy.

Light Cavalry

Within the Light Cavalry would be three major types of light cavalry, the Chasseurs a Cheval, the Hussars, and the Cheval-Legere Lancier (or Lancers). All of light cavalry had two purposes, scouting and harassing, of which they all excelled in. The latter would require a special curved sword that would be excellent on riding men down as they were running but not as good on a standard engagement, the reason being that curved swords would cut rather than stab, thus a shallow and non-lethal wound would appear.

Medium Cavalry

In most European armies, the Dragoon would have been considered heavy cavalry but in France it holds an odd place. Dragoons were normally used as cavalry that was heavier than light but not as special or large as the heavy cavalry. Thus they typically used a standard straight sword that would be useful for stabbing. Due to the nature of dragoons being somewhat heavy but not light cavalry, they were typically used as heavy cavalry in smaller corps or to help stiffen light cavalry and give them more combat power.

Heavy Cavalry

Here, there would be three more types of heavy cavalry; the carabineer a cheval, the Cuirassier, and the Grenadier a Cheval de la Garde with the first two as major combat elements. These would be large men on large horses with large straight swords. The purposes of these units would be as almost literal battering rams, large heavy men on large heavy horses charging into an enemy battalion or regiment, using the weight of them on the horse, and driving their large straight swords into the enemy. This would break the enemy and cause an opening within the enemy's line.

And while I apologize for going into detail about cavalry usage, the usage and the weapon are carefully tied together. Napoleon would often get after the Grenadiers a Cheval, yelling at them to not slash but to stab.

The British Army first standardized cavalry sabre designs in 1788.

(Before this, in early times, soldiers supplied their own swords, later, swords might be supplied by the regiment, but each regiment might use slightly different designs (or even issue multiple sabre designs to their troopers)).

In 1788, the British standardized on two cavalry sabres; A straight, thrusting sabre for heavy cavalry, and a curved, slashing sabre for light cavalry.

The debate between the thrust and the cut and the straight versus curved sabre went on until the straight, 1908 pattern cavalry sabre (thought by many swordsmen to be the best cavalry sabre ever made) was introduced.

The standard British sabres were redesigned in 1796, again, straight for the Heavies, curved for the Lights. The British light cavalry sabre was essentially copied by the Prussian cavalry.

They were redesigned again in 1821. Both were slightly curved this time (which pleased no one).

In 1853, a new pattern sabre was introduced. This time one sabre for both the heavy and light cavalry. It had a very slight curve. This sabre was used extensively by both US and Confederate cavalry in the American Civil War. It was not loved by the British cavalry, who saw it as a bit of a bastard compromise.

In 1864, this sabre acquired a basket hilt.

The next re-design was in 1885. Again, a slightly curved sword. The sabre was too short to allow a cavalryman to strike an enemy lying on the ground. This was corrected in the 1899 pattern, with a longer blade.

Finally, in 1908, the British cavalry went back to a straight bladed sabre, with a basket hilt and a semi pistol grip handle. This was a superb thrusting sabre. The 1912 Officer’s pattern was a fancier version of the same sword, and the 1913 Patton sabre, adopted by the US cavalry was very similar.

Basically, cavalry sabre designers needed to decide whether to optimize the sword for cut or thrust, or design a hybrid that could do both (but not as well). Generally the troopers seemed to prefer the optimized weapons, though the designers spent a long time trying to produce compromises. The British cavalry and the US eventually settled on a straight weapon, optimized for the thrust.

All European cavalries had this same dilemma, and most armies had very similar sabres to British sabres at one time or another.

http://www.katetattersall.com/british-cavalry-sabres-of-the-early-to-mid-1800s/

http://www.militaryheritage.com/swords2.htm

https://en.wikipedia.org/wiki/Pattern_1908_and_1912_cavalry_swords